Unyielding Flame socketing question.

Do support gems in this item support the Commandment of Inferno that triggers from it?

While we're here, same question for Lab enchants on gloves. Might have different answers since the gloves may not be considered as coming from the base item?

I think PoB answered this for me, seems like a no, if the program is right.

Still, this item seems insane, even if my ES takes a major hit for it.